User equipment (UE) detection is an important ingredient of any uplink random access system. Compressed sensing (CS) is a tool for UE detection in scenarios where UE activity is sparse in the sense that from a large number of UEs that could attempt to access the system at a given instant, typically a relatively small number are in fact attempting to access the system. In other words, a small number of UEs among a large pool of UEs are simultaneously active.
A standard CS problem involves solving an underdetermined set of (noisy or noiseless) equations in terms of an unknown sparse vector based on a number of observations. The number of nonzero elements in the unknown vector is much less than the number of observations. The set of equations can be expressed as follows:yM×1=PM×N·hN×1+nM×1 where the elements of the equation are:
yM×1=is a set of M observations;
hN×1 is a set of N unknowns;
PM×N is a matrix that defines linear combinations of the unknowns;
nM×1 is a set of noise components.
with K«M<N, where K is the number of nonzero elements of h.
The CS problem is usually cast as an optimization problem. One typical example is the following convex optimization problem
      h    ^    =                                          arg            ⁢                                                  ⁢            min                    ⁢                                                h            ⁢              1        2            ⁢                                              y            -            Ph                                    2        2              +          λ      ⁢                                  h                          1            ∥·∥2 and ∥·∥1 denote I2-norm and I1-norm of a vector, respectively, defined as∥x∥2=√{square root over (Σi|xi|2)}and ∥x∥1=Σi|xi∥
The UE detection problem can be recast as a CS problem based on the following set of equations:yM×1=PM×N·hN×1+nM×1 where the elements of the equation are:                yM×1=is a set of M observations;        hN×1 is a set of N unknowns representing the vector of channel coefficients of the UEs; each active UE corresponds to a nonzero element in h and each inactive UE corresponds to a zero element in h. It is noted that h is not the channel per se, but is a CS determined single value that represents a fixed channel over the entire pilot sequence; following CS detection, h is the CS detection output;        PM×N is a pilot matrix;        nM×1 is a set of noise components;with K<<M<N, where        
K is the number of nonzero elements of h, which is equal to the number of UEs that are in fact active at a given instant;
N is the size of the pool of UEs that might be active;
M is the number of observations.